In the limit of ultrashort probe pulses, i.e., \(\sigma _{\rm pr}\)  smaller than any nonequilibrium physical timescales of the system, the envelope \(g(\tau ;t)^2\)  can be approximated by a delta function \(\delta (\tau -t)\) , leading to an explicit solution consistent with the equilibrium sum-rule integral [1]}. However, when the probe pulse has a finite time duration (as in spectroscopy experiments with high energy resolution), this approximation breaks down. In order to extract the instantaneous QFI from this implicit equation Eq. (REF ), we expand its left-hand side and convert the equation into a self-consistent integro-differential problem. As detailed in Appendix , this leads to
\(f_\mathrm {Q}(q, t) = 8\sigma _\mathrm {pr} \sqrt{\pi } \int \displaylimits _{-\infty }^{+\infty } d\omega \, S( q, \omega , t) + \sum _{m=1}^\infty \frac{\mathcal {C}_m}{2m!} \frac{\partial ^{2m}f_\mathrm {Q}}{\partial t^{2m}},\) 
